Finite Group Databasebeta

← #37532804

Subgroups — #37532804

Order 23168.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041653C211yesmaximal, characteristic, complemented
2#26041474C212yesmaximal, complemented
3#26041028C211yesmaximal, characteristic, complemented
4D4⋊#13089678C212yesmaximal, complemented
5#555×C181C211yesmaximal, characteristic, complemented
6C2×#15668436—22no—
7C2×#15668434—22no—
8#19153813C2211yescharacteristic
9C4⋊#13089678C2212yes—
10#19153736—22no—
11C4×#13089678C2211yescharacteristic
12C4⋊Q8×C181C2211yescharacteristic
13#124×C181—22no—
14#121×C181—22no—
15C4×D4×C181C2212yes—
16#117×C181—22no—
17C2×D724—22no—
18C2×D724—21no—
19#15668436—44no—
20#15668434—44no—
21C2×#13089678—21no—
22C2×#13089678—22no—
23C2×Q8×C181D412yes—
24C2×D4×C181—22no—
25C2×D4×C181D412yes—
26C22×C4×C181—22no—
27C4⋊C4×C181—22no—
28C4⋊C4×C181D412yes—
29C22⋊C4×C181—24no—
30C22⋊C4×C181—22no—
31C42×C181—22no—
32C42×C181C2311yescharacteristic
33C22×D181—41no—
34D724—42no—
35D724—42no—
36D724—42no—
37#13089678—22no—
38#13089678—42no—
39C23×C181—22no—
40Q8×C181—42no—
41D4×C181—22no—
42D4×C181—22no—
43D4×C181—82no—
44C2×C4×C181—22no—
45C2×C4×C181—22no—
46C2×C4×C181—42no—
47C2×C4×C181—42no—
48C2×C4×C181C2×D411yescharacteristic
49C2×C4×C181C2×D412yes—
50C22×C181—42no—
51C22×C181—42no—
52C22×C181—42no—
53C22×C181#12011yescharacteristic
54C2×D181—82no—
55C2×D181—81no—
56C4×C181—21no—
57C4×C181—42no—
58C4×C181—42no—
59C4×C181#13512yes—
60C4×C181AGL(1,ℤ/8)12yes—
61C2×C181—82no—
62C2×C181#44912yes—
63C2×C181#45311yescharacteristic
64D181—161no—
65C181#162411yescharacteristic, complemented, Sylow
66#1624—1811nomaximal, Sylow
67#555D18111yescharacteristic, complemented
68#492—1811no—
69#459—1812no—
70#333—1811no—
71D4⋊C8—1812no—
72C2×SD16—3622no—
73C2×D8—3622no—
74C4⋊Q8C2×D18111yescharacteristic
75#127—1811no—
76#124—22no—
77#121—22no—
78C4×D4C2×D18112yes—
79#117—22no—
80C4⋊C8—1812no—
81#102—3622no—
82C4×C8—1811no—
83C2×Q8C22⋊D18112yes—
84C2×D4—22no—
85C2×D4—3622no—
86C2×D4—3621no—
87C2×D4C22⋊D18112yes—
88C22×C4—22no—
89SD16—7244no—
90D8—7244no—
91C2×C8—3621no—
92C2×C8—3622no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D18112yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D18111yescharacteristic
99C23—22no—
100C23—7241no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—7242no—
106D4—7242no—
107D4—7242no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D18111yescharacteristic
113C2×C4C2×C22⋊D18112yes—
114C8—3622no—
115C8—7242no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—14482no—
120C22—14481no—
121C22#1915378511yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#1915383612yes—
126C4#1915384112yes—
127C2—82no—
128C2—28961no—
129C2#2604142412yes—
130C2#2604144711yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#555×C181C2
socleC22×C181#120
hypercenter#555D181
hypocenterC181#1624
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26041653C2semidirect
2. #26041474C2semidirect
3. #26041028C2semidirect
4. D4⋊#13089678C2semidirect
5. #555×C181C2semidirect
65. C181#1624semidirect
67. #555D181semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260416532
#260414742
#260410282
D4⋊#130896782
#555×C1812
#1624181